Saturation, torsion, and contorsion: Difference between revisions
→Torsion and contorsion: A contorted EDO may do a relatively good job of representing a JI lattice, but this is not guaranteed |
Rework the structure of this article to improve beginner friendliness. These concepts should not be discussed one by one, but should be grouped together and separated by topics one might be interested in: the math, the algorithms, and the terminology |
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{{Beginner|Mathematical theory of saturation}} | {{Beginner|Mathematical theory of saturation}} | ||
In [[regular temperament theory]], a [[mapping]] or [[comma basis]] is '''saturated''' if it represents a temperament without any redundancies due to a common factor, such that the temperament is represented with the simplest matrix. '''Torsion''' and '''contorsion''' are the opposite of saturation, but torsion only regards comma bases, whereas contorsion only regards mappings. In other words, a comma basis is either saturated, or it ''has torsion'', and a mapping is either saturated, or it ''has contorsion'' (or ''is contorted''). The greatest factor of a mapping has been called its '''contorted order'''. | |||
In general, being unsaturated is a bad thing<ref>Technically speaking, saturation is a property of lattices, not the matrices that generate them, and is only "bad" when referring to a comma basis or a lattice of supporting maps.</ref>, as the redundancy means that the same temperament information can be represented in a simpler way. There are other manners in which unsaturation is bad, and these depend on whether the matrix is a comma basis or a mapping, discussed below. For all these reasons, unsaturated matrices are typically considered to not truly represent temperaments. | |||
A comma basis with torsion is rarely useful at all. It states that a power of a ratio is made to [[vanish]] but does not explicitly state that the ratio itself is made to vanish. For instance, (81/80)<sup>2</sup> is made to vanish but 81/80 is not. From a mathematical standpoint, there are multiple ways to interpret this situation. Historically, a group-theory formalism was used, leading to the interpretation that the ratio itself was indeed ''not'' made to vanish, despite this being musically absurd. Using a linear-algebra formalism as is preferred now, however, no such absurdity is suggested. This historical usage of the group theory formalism is another reason why it is bad practice to use comma bases with torsion. | |||
Torsion also occurs in a similar situation where the list of commas defines a [[periodicity block]]; in fact, this was its original use case. | |||
Contorted mappings represent a different sort of pathology from comma bases with torsion: they involve tuning systems for which some pitches are unmapped, i.e. no [[just intonation]] interval maps to them. Contorted mappings can be useful in a way that unsaturated comma bases cannot: these matrices do at least represent temperament-like systems with sensible notions of pitch. When compared to the temperament that is represented by the saturated version of the same mapping, they simply have these extra unmapped pitches that no just ratio tempers to. | |||
In practical terms, a contorted edo may do a relatively good job of approximating a given JI lattice, but it does not actually use all notes from the tuning, instead looping back to 1/1 before using up all the notes. For example, 24edo is not too bad at approximating the 5-limit, but it only uses its 12edo subset to do that, since the odd degrees of 24edo fall outside of the 5-limit approximation. | |||
For a more detailed discussion on these issues, see [[Pathology of enfactoring|Pathology of saturation]]. | |||
== | == Mathematical definition == | ||
A [[mapping]] is saturated when no common factor is found in its rows (i.e. generator maps). A [[comma basis]], the dual of a mapping, is saturated when no common factor is found in its columns (i.e. comma vectors). | |||
To be more specific, a mapping is saturated if no {{w|coprime integers|coprime}}<ref>If the multiples used on the linear combinations themselves have a GCD > 1, the resulting row will always have a GCD > 1, and such a linear combination therefore can not be used to demonstrate unsaturation. For example, consider the matrix {{mapping| 1 0 -4 | 0 1 4 }}; we can find the linear combination of rows {{nowrap| 2 × {{val| 1 0 -4 }} + 4 × {{val| 0 1 4 }} {{=}} {{val| 2 4 8 }}}}, which has a GCD of 2, but that is clearly a result of the fact that we used 2× and 4× of the original rows, and 2 and 4 have a GCD of 2. So this linear combination does not prove that the matrix is unsaturated. It is still possible that another linear combination might prove it, but this one does not.</ref> integer {{w|linear combination}} of its rows can produce another row whose entries have a common factor other than 1. For example, {{mapping| 3 0 -1 | 0 3 5 }} is ''not'' saturated, because {{nowrap|{{val| 3 0 -1 }} − {{val| 0 3 5 }} {{=}} {{val| 3 -3 6 }}}}, which has a common factor of 3. A mapping which consists of a single row with a common factor, such as {{mapping| 24 38 56 }} with a visible common factor of 2, is also not saturated. | |||
The greatest factor possible to create as a GCD of a coprime linear combination of rows in this way is called the mapping's '''greatest factor'''<ref>This term is inspired by H. J. S. Smith's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor".</ref>, and so an alternative definition of saturation would be that the mapping must have a greatest factor of 1. | |||
== Saturation algorithms == | |||
An unsaturated mapping or comma basis can be made saturated, ensuring our ability to most simply—and thereby uniquely—identify temperaments using only matrices. This need can otherwise be satisfied using [[wedgie]]s. The simplest and fastest algorithm for saturating matrices is called [[column Hermite defactoring]]. For more information on such algorithms, see [[Defactoring algorithms|Saturation algorithms]]. | |||
The term was | == Terminology == | ||
The term ''saturation'' was coined by {{w|Nicolas Bourbaki}} in 1972<ref>[https://pdfcoffee.com/commutative-algebra-bourbaki-pdf-free.html Nicolas Bourbaki. ''Commutative Algebra'']</ref>, working in the field of commutative algebra. It came to RTT via [[Gene Ward Smith]] and [[Graham Breed]]'s observations of the work of the mathematician {{w|William A. Stein|William Stein}} and his {{w|SageMath|Sage}} software<ref>It may also have come through PARI/GT<!-- typo of PARI/GP? -->.</ref>. The earliest identified terminology for this concept was in 1861 by {{w|Henry John Stephen Smith|H. J. S. Smith}}<ref>H. J. S. Smith is the creator of the {{w|Smith normal form}} used in [[Defactoring algorithms #Precedent: Smith defactoring|Gene Ward Smith's saturation algorithm]].</ref> who called saturated matrices "prime matrices"<ref>Also from ''On Systems of Linear Indeterminate Equations and Congruences'', linked above. Neither ''prime matrix'' nor ''greatest divisor'' seems to have caught on in the mathematical community.</ref>. | |||
''''' | The term ''torsion'' has been used since at least as early as 1932<ref>[https://scholar.google.com/scholar?q=%22torsion+group%22&hl=en&as_sdt=0%2C5&as_ylo=1900&as_yhi=1940 Google Scholar: Torsion group]</ref><ref>[https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from Stack Exchange | ''Where does the word "torsion" in algebra come from?'']</ref> and came to RTT from the mathematical field of group theory. The term ''contorsion'' was invented for RTT in 2002 by [[Paul Erlich]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456 Yahoo! Tuning Group | ''My top 5--for Paul'']</ref>, as a play on the word "co-torsion", being dual to the situation with "torsion" above. | ||
[[Dave Keenan]] and [[Douglas Blumeyer]] have proposed '''''defactoring''''' as a replacement for ''saturation'', and '''''enfactoring''''' as a replacement for both ''torsion'' and ''contorsion''. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion). These terms are used in their writings. See [[Defactoring terminology proposal]] for details. | |||
== References and footnotes == | == References and footnotes == | ||
<references /> | <references/> | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Math]] | [[Category:Math]] | ||
Revision as of 09:15, 17 February 2025
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This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily. The corresponding expert page for this topic is Mathematical theory of saturation. |
In regular temperament theory, a mapping or comma basis is saturated if it represents a temperament without any redundancies due to a common factor, such that the temperament is represented with the simplest matrix. Torsion and contorsion are the opposite of saturation, but torsion only regards comma bases, whereas contorsion only regards mappings. In other words, a comma basis is either saturated, or it has torsion, and a mapping is either saturated, or it has contorsion (or is contorted). The greatest factor of a mapping has been called its contorted order.
In general, being unsaturated is a bad thing[1], as the redundancy means that the same temperament information can be represented in a simpler way. There are other manners in which unsaturation is bad, and these depend on whether the matrix is a comma basis or a mapping, discussed below. For all these reasons, unsaturated matrices are typically considered to not truly represent temperaments.
A comma basis with torsion is rarely useful at all. It states that a power of a ratio is made to vanish but does not explicitly state that the ratio itself is made to vanish. For instance, (81/80)2 is made to vanish but 81/80 is not. From a mathematical standpoint, there are multiple ways to interpret this situation. Historically, a group-theory formalism was used, leading to the interpretation that the ratio itself was indeed not made to vanish, despite this being musically absurd. Using a linear-algebra formalism as is preferred now, however, no such absurdity is suggested. This historical usage of the group theory formalism is another reason why it is bad practice to use comma bases with torsion.
Torsion also occurs in a similar situation where the list of commas defines a periodicity block; in fact, this was its original use case.
Contorted mappings represent a different sort of pathology from comma bases with torsion: they involve tuning systems for which some pitches are unmapped, i.e. no just intonation interval maps to them. Contorted mappings can be useful in a way that unsaturated comma bases cannot: these matrices do at least represent temperament-like systems with sensible notions of pitch. When compared to the temperament that is represented by the saturated version of the same mapping, they simply have these extra unmapped pitches that no just ratio tempers to.
In practical terms, a contorted edo may do a relatively good job of approximating a given JI lattice, but it does not actually use all notes from the tuning, instead looping back to 1/1 before using up all the notes. For example, 24edo is not too bad at approximating the 5-limit, but it only uses its 12edo subset to do that, since the odd degrees of 24edo fall outside of the 5-limit approximation.
For a more detailed discussion on these issues, see Pathology of saturation.
Mathematical definition
A mapping is saturated when no common factor is found in its rows (i.e. generator maps). A comma basis, the dual of a mapping, is saturated when no common factor is found in its columns (i.e. comma vectors).
To be more specific, a mapping is saturated if no coprime[2] integer linear combination of its rows can produce another row whose entries have a common factor other than 1. For example, [⟨3 0 -1], ⟨0 3 5]] is not saturated, because ⟨3 0 -1] − ⟨0 3 5] = ⟨3 -3 6], which has a common factor of 3. A mapping which consists of a single row with a common factor, such as [⟨24 38 56]] with a visible common factor of 2, is also not saturated.
The greatest factor possible to create as a GCD of a coprime linear combination of rows in this way is called the mapping's greatest factor[3], and so an alternative definition of saturation would be that the mapping must have a greatest factor of 1.
Saturation algorithms
An unsaturated mapping or comma basis can be made saturated, ensuring our ability to most simply—and thereby uniquely—identify temperaments using only matrices. This need can otherwise be satisfied using wedgies. The simplest and fastest algorithm for saturating matrices is called column Hermite defactoring. For more information on such algorithms, see Saturation algorithms.
Terminology
The term saturation was coined by Nicolas Bourbaki in 1972[4], working in the field of commutative algebra. It came to RTT via Gene Ward Smith and Graham Breed's observations of the work of the mathematician William Stein and his Sage software[5]. The earliest identified terminology for this concept was in 1861 by H. J. S. Smith[6] who called saturated matrices "prime matrices"[7].
The term torsion has been used since at least as early as 1932[8][9] and came to RTT from the mathematical field of group theory. The term contorsion was invented for RTT in 2002 by Paul Erlich[10], as a play on the word "co-torsion", being dual to the situation with "torsion" above.
Dave Keenan and Douglas Blumeyer have proposed defactoring as a replacement for saturation, and enfactoring as a replacement for both torsion and contorsion. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion). These terms are used in their writings. See Defactoring terminology proposal for details.
References and footnotes
- ↑ Technically speaking, saturation is a property of lattices, not the matrices that generate them, and is only "bad" when referring to a comma basis or a lattice of supporting maps.
- ↑ If the multiples used on the linear combinations themselves have a GCD > 1, the resulting row will always have a GCD > 1, and such a linear combination therefore can not be used to demonstrate unsaturation. For example, consider the matrix [⟨1 0 -4], ⟨0 1 4]]; we can find the linear combination of rows 2 × ⟨1 0 -4] + 4 × ⟨0 1 4] = ⟨2 4 8], which has a GCD of 2, but that is clearly a result of the fact that we used 2× and 4× of the original rows, and 2 and 4 have a GCD of 2. So this linear combination does not prove that the matrix is unsaturated. It is still possible that another linear combination might prove it, but this one does not.
- ↑ This term is inspired by H. J. S. Smith's On Systems of Linear Indeterminate Equations and Congruences, where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor".
- ↑ Nicolas Bourbaki. Commutative Algebra
- ↑ It may also have come through PARI/GT.
- ↑ H. J. S. Smith is the creator of the Smith normal form used in Gene Ward Smith's saturation algorithm.
- ↑ Also from On Systems of Linear Indeterminate Equations and Congruences, linked above. Neither prime matrix nor greatest divisor seems to have caught on in the mathematical community.
- ↑ Google Scholar: Torsion group
- ↑ Stack Exchange | Where does the word "torsion" in algebra come from?
- ↑ Yahoo! Tuning Group | My top 5--for Paul